In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.
Notation Let w {\displaystyle w} be a Fourier series with Fourier coefficients c k {\displaystyle c_{k}} , relating to each other as
w ( θ ) = ∑ k = − ∞ ∞ c k e i k θ , θ ∈ [ 0 , 2 π ] , {\displaystyle w(\theta )=\sum _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\qquad \theta \in [0,2\pi ],}
c k = 1 2 π ∫ 0 2 π w ( θ ) e − i k θ d θ , {\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }w(\theta )e^{-ik\theta }\,d\theta ,}
such that the n × n {\displaystyle n\times n} Toeplitz matrices T n ( w ) = ( c k − l ) 0 ≤ k , l ≤ n − 1 {\displaystyle T_{n}(w)=\left(c_{k-l}\right)_{0\leq k,l\leq n-1}} are Hermitian, i.e., T n ( w ) = T n ( w ) ∗ {\displaystyle T_{n}(w)=T_{n}(w)^{\ast }} , or equivalently c − k = c k ¯ {\displaystyle c_{-k}={\overline {c_{k}}}} . Then both w {\displaystyle w} and the eigenvalues ( λ m ( n ) ) 0 ≤ m ≤ n − 1 {\displaystyle (\lambda _{m}^{(n)})_{0\leq m\leq n-1}} of T n ( w ) {\displaystyle T_{n}(w)} are real-valued and the determinant of T n ( w ) {\displaystyle T_{n}(w)} is given by
det T n ( w ) = ∏ m = 1 n − 1 λ m ( n ) {\displaystyle \det T_{n}(w)=\prod _{m=1}^{n-1}\lambda _{m}^{(n)}} .
Szegő theorem Under suitable assumptions the Szegő theorem states that
lim n → ∞ 1 n ∑ m = 0 n − 1 F ( λ m ( n ) ) = 1 2 π ∫ 0 2 π F ( w ( θ ) ) d θ {\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{m=0}^{n-1}F(\lambda _{m}^{(n)})={\frac {1}{2\pi }}\int _{0}^{2\pi }F(w(\theta ))\,d\theta }
for any function F {\displaystyle F} that is continuous on the range of w {\displaystyle w} . In particular
i.e. the arithmetic mean of λ ( n ) {\displaystyle \lambda ^{(n)}} converges to the integral of w {\displaystyle w} , i.e. c 0 {\displaystyle c_{0}} .
First Szegő theorem The first Szegő theorem states that, if right-hand side of (1) holds and w ≥ 0 {\displaystyle w\geq 0} , then
holds for w > 0 {\displaystyle w>0} and w ∈ L 1 {\displaystyle w\in L^{1}} . The RHS of (2) is the geometric mean of w {\displaystyle w} (well-defined by the arithmetic-geometric mean inequality).
Second Szegő theorem Let c ^ k {\displaystyle {\widehat {c}}_{k}} be the Fourier coefficient of log w ∈ L 1 {\displaystyle \log w\in L^{1}} , written as
c ^ k = 1 2 π ∫ 0 2 π log ( w ( θ ) ) e − i k θ d θ {\displaystyle {\widehat {c}}_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }\log(w(\theta ))e^{-ik\theta }\,d\theta }
The second (or strong) Szegő theorem states that, if w ≥ 0 {\displaystyle w\geq 0} , then
lim n → ∞ det T n ( w ) e ( n + 1 ) c ^ 0 = exp ( ∑ k = 1 ∞ k | c ^ k | 2 ) . {\displaystyle \lim _{n\to \infty }{\frac {\det T_{n}(w)}{e^{(n+1){\widehat {c}}_{0}}}}=\exp \left(\sum _{k=1}^{\infty }k\left|{\widehat {c}}_{k}\right|^{2}\right).}
See also Trigonometric moment problem Verblunsky's theorem
References